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User blog:Holomanga/Metric Spaces 2: It's Time..
What is it time for? No idea. Convergence, Binch Okay, it's time for convergence, as that title says. Why is it time for convergence? The answer is simple. The first question of your homework is on it. I assume I am violating 300 copyright laws by doing this. If you work at the Mathematical Insitute in Oxford University, please don't sue me. Okay. So, um. Um. The question is basically that the sequence xn converges to the point x, and it also converges to the point x'. We're meant to discover that these two points are actually the same point. A sequence can't converge to two different points at the same time. 'PAUSE FOR A MOMENT AND TRY IT YOURSELF. YOU'LL FEEL VERY SMART. ' Let's look at the definition of convergence again. Convergence: for the sequence xn over the set X with a metric dX, xn converges to a if for any \epsilon>0 , no matter how small, you can find an N such that for all n>N there is d_X (x_n,a)<\epsilon . Ah, I see how we can do this. Pick an N that's big enough that the distance between d(xn, x) < ε and the distance between d(xn, x') < ε. By the triangle inequality, d(x, x') < 2ε. But we know that ε > 0, so 2ε > 0. The only number that is always less than something that is greater than 0 is 0. So d(x, x') is 0, so x = x'. I'm dabbing. I am actually dabbing right now. This is maths. I just did maths. This is the first time I've ever done real maths. This is the content I crave. I felt so smart doing that. I want to keep doing this forever. Open Sets? So, here's another question. Question one, part two of the problem sheet I was working from. The issue with this is that I have no idea what closed or open sets actually are. I know they're a big deal, because I've saw them show up in a bunch of places before, so let's get cracking with working out what they mean. Basically, if you take a subset of a metric space (just collect a bunch of points, but keep the same metric), some of them are either open or closed. This is an important distinction to make because mathematicians are weird. Also they're a really big deal. Like a freakishly big deal. Have you saw how much actually depends on open and closed sets? Whatever they are, you really ought to care about them, because with them you can apparently control the universe. My notes start geometrically. Let's just see where they go with it. In the metric space of set X equipped with the metric d, * The open ball of radius ε about x0 is the set B(x_0, \epsilon) = \{ x \in X : d(x, x_0) < \epsilon \} * The closed ball of radius ε about x0 is the set \overline{B} (x_0, \epsilon) = \{ x \in X : d(x, x_0) \leq \epsilon \} In other words, open balls are sets that contain all the points that have a distance of less than ε from their central point as defined by the metric, and closed balls also contain all the point that have a distance of exactly ε from the center. These are called balls because in \mathbb{R}^3 with the metric d_2 you know and love, they look like balls. Now, um, bear with me for a moment, but there's this definition in my notes and it says that it follows clearly from the definition but I really don't see it. The function f : X → Y is continuous at a if, and only if, for any open ball B(f(a), \epsilon) centred at f(a) there is an open ball B(a, \delta) such that f(B(a, \delta)) \subseteq B(f(a), \delta) This is another defintion of continuity - to be continuous is to hold this property. It says that it's clear from the definition of continuity, so let me just go back and check that definition (and from open balls, which we have right here). Continuity: A function f from the set X, with a metric dX, to the set Y, with a metric dY is continuous for any \epsilon >0 , no matter how small, you can find a \delta >0 such that d_X (x,a)<\delta and d_Y (f(x),f(a))<\epsilon , for every a. Ah, I see. Because d(a, x_0) < \delta , that means that x0 must be a point inside the ball B(a, \delta) , and because d(f(a), f(x_0)) < \epsilon that means that f(x0) must be a point inside the ball B(f(a), \epsilon) . But we know that x0 is inside B(a, \delta) , which means that, drum-motherfuckin'-roll All points f(B(a, \delta)) are inside the ball B(f(a), \epsilon) . And hey, that's the lemma we started with. Nice. We did it. This is a real victory for maths and for us. Next Up There's this really exciting section that has the word TOPOLOGY in it. Like the actual word, with those letters, in that order. I can't wait. Skimread suggests that it's closely linked to this new open ball definition of continuity. Category:Blog posts